How unprovable is Rabin's decidability theorem?

We study the strength of set-theoretic axioms needed to prove Rabin's theorem on the decidability of the MSO theory of the infinite binary tree. We first show that the complementation theorem for tree automata, which forms the technical core of typical proofs of Rabin's theorem, is equivalent over the moderately strong second-order arithmetic theory $\mathsf{ACA}_0$ to a determinacy principle implied by the positional determinacy of all parity games and implying the determinacy of all Gale-Stewart games given by boolean combinations of ${\bf \Sigma^0_2}$ sets. It follows that complementation for tree automata is provable from $\Pi^1_3$- but not $\Delta^1_3$-comprehension. We then use results due to MedSalem-Tanaka, M\"ollerfeld and Heinatsch-M\"ollerfeld to prove that over $\Pi^1_2$-comprehension, the complementation theorem for tree automata, decidability of the MSO theory of the infinite binary tree, positional determinacy of parity games and determinacy of $\mathrm{Bool}({\bf \Sigma^0_2})$ Gale-Stewart games are all equivalent. Moreover, these statements are equivalent to the $\Pi^1_3$-reflection principle for $\Pi^1_2$-comprehension. It follows in particular that Rabin's decidability theorem is not provable in $\Delta^1_3$-comprehension.Comment: 21 page

From winning strategy to Nash equilibrium

Game theory is usually considered applied mathematics, but a few game-theoretic results, such as Borel determinacy, were developed by mathematicians for mathematics in a broad sense. These results usually state determinacy, i.e. the existence of a winning strategy in games that involve two players and two outcomes saying who wins. In a multi-outcome setting, the notion of winning strategy is irrelevant yet usually replaced faithfully with the notion of (pure) Nash equilibrium. This article shows that every determinacy result over an arbitrary game structure, e.g. a tree, is transferable into existence of multi-outcome (pure) Nash equilibrium over the same game structure. The equilibrium-transfer theorem requires cardinal or order-theoretic conditions on the strategy sets and the preferences, respectively, whereas counter-examples show that every requirement is relevant, albeit possibly improvable. When the outcomes are finitely many, the proof provides an algorithm computing a Nash equilibrium without significant complexity loss compared to the two-outcome case. As examples of application, this article generalises Borel determinacy, positional determinacy of parity games, and finite-memory determinacy of Muller games

A Functional (Monadic) Second-Order Theory of Infinite Trees

This paper presents a complete axiomatization of Monadic Second-Order Logic (MSO) over infinite trees. MSO on infinite trees is a rich system, and its decidability ("Rabin's Tree Theorem") is one of the most powerful known results concerning the decidability of logics. By a complete axiomatization we mean a complete deduction system with a polynomial-time recognizable set of axioms. By naive enumeration of formal derivations, this formally gives a proof of Rabin's Tree Theorem. The deduction system consists of the usual rules for second-order logic seen as two-sorted first-order logic, together with the natural adaptation In addition, it contains an axiom scheme expressing the (positional) determinacy of certain parity games. The main difficulty resides in the limited expressive power of the language of MSO. We actually devise an extension of MSO, called Functional (Monadic) Second-Order Logic (FSO), which allows us to uniformly manipulate (hereditarily) finite sets and corresponding labeled trees, and whose language allows for higher abstraction than that of MSO

Quantitative games with interval objectives

Traditionally quantitative games such as mean-payoff games and discount sum games have two players -- one trying to maximize the payoff, the other trying to minimize it. The associated decision problem, "Can Eve (the maximizer) achieve, for example, a positive payoff?" can be thought of as one player trying to attain a payoff in the interval $(0,\infty)$. In this paper we consider the more general problem of determining if a player can attain a payoff in a finite union of arbitrary intervals for various payoff functions (liminf, mean-payoff, discount sum, total sum). In particular this includes the interesting exact-value problem, "Can Eve achieve a payoff of exactly (e.g.) 0?"Comment: Full version of CONCUR submissio

A Comparison of BDD-Based Parity Game Solvers

Parity games are two player games with omega-winning conditions, played on finite graphs. Such games play an important role in verification, satisfiability and synthesis. It is therefore important to identify algorithms that can efficiently deal with large games that arise from such applications. In this paper, we describe our experiments with BDD-based implementations of four parity game solving algorithms, viz. Zielonka's recursive algorithm, the more recent Priority Promotion algorithm, the Fixpoint-Iteration algorithm and the automata based APT algorithm. We compare their performance on several types of random games and on a number of cases taken from the Keiren benchmark set.Comment: In Proceedings GandALF 2018, arXiv:1809.0241

無限ゲームと様相μ計算の部分体系についてのオートマトン理論的研究

要約のみTohoku University田中一之課

Emptiness of Zero Automata Is Decidable

Zero automata are a probabilistic extension of parity automata on infinite trees. The satisfiability of a certain probabilistic variant of MSO, called TMSO+zero, reduces to the emptiness problem for zero automata. We introduce a variant of zero automata called nonzero automata. We prove that for every zero automaton there is an equivalent nonzero automaton of quadratic size and the emptiness problem of nonzero automata is decidable, with complexity co-NP. These results imply that TMSO+zero has decidable satisfiability